The mod and div Functions in .NET

Develop Data Matrix in .NET The mod and div Functions
15.3 The mod and div Functions
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In Section 15.1, we showed that (15.1) defines at most one remainder r and quotient d for all integers P and all strictly positive natural numbers Q. The con-
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15: Remainder Computation struction of the elementary algorithm in Section 15.2 (for both the cases that 0 ^ P and P ^ 0) establishes that there is at least one remainder r and quotient d for all integers P and all strictly positive natural numbers Q. Together, we have thus established that there is a function that maps given integers P and Q, where 0 < Q, to the remainder, r, and quotient, d, after dividing P by Q, as specified formally by (15.1). A specification (a relation between input and output values) has a functional solution if there is exactly one output for each input. Exhibiting an algorithm, no matter how efficient or inefficient, satisfying the specification is the most effective way of demonstrating the existence of at least one solution to a specification. Recognizing that a specification has a functional solution has important consequences for reasoning about the specification. The consequences are realized by naming the function and expressing the functionality by a simple equivalence. (Some logic texts refer to this process as Skolemization.) In the case of the specification (15.1), there are two quantities involved, so we give separate names to each. The standard name in mathematics for the remainder function is 'mod'; the symbol '-r' is used for the quotient. Given integers P and Q, where 0 < Q, P mod Q denotes the remainder, and P + Q denotes the quotient, after dividing P by Q. The existence and uniqueness of these functions is expressed by the calculational rule: r = PmodQ A d = P - Q = 0 ^ r < Q A P = Qxd + r . (15.6)
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Splitting the equivalence into a mutual implication, the '=>' expresses the existence of a solution to (15.1), and the '<=' expresses the uniqueness of a solution to (15.1). We established the existence of a solution by constructing the elementary algorithm of Section 15.2. The uniqueness of the solution was established in Section 15.1. In 6, the symbol' V was used for integer division, but a different definition was given. We show in Section 15.3.3 that the two definitions are equivalent. For the moment, however, we take care to use only properties of -r that are derived from (15.6). The standard convention in mathematics texts is that mod, like -5-, has higher precedence than addition but lower precedence than multiplication. So, for example, m + nmodQ and m + (nmodQ) are equal. Also, raxnmodQ and (raxn) mod Q are equal. Giving multiplication precedence over mod is undoubtedly due to the fact that it is common to denote multiplication by juxtaposition, and the eye naturally groups ra and n together in mnmodQ. When multiplication is explicitly denoted, the algebraic properties make it undesirable to give multiplication precedence over mod. Our own preference is therefore for the opposite convention. However, to avoid confusion with other texts, we take the middle road of including parentheses, even though some may be omitted.
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15.3 The mod and div Functions
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15.3.1 Basic Properties
Here is how to use (15.6) in a few simple cases. First, we make the left side of the equivalence trivially true by instantiating r to P mod Q and d to F ~ Q. We get 0 ^ PmodQ <Q A P = Q x ( P ^ Q ) + PmodQ . (15.7) Next, we make the second and third conjuncts true by instantiating r to 0 and d to P -r Q. We get 0 = PmodQ = P = Qx(P-Q) . Now, we instantiate r to P mod Q and d to 0. We get true { = = (15.6)withr,d := P m o d Q . O } (15.8)
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